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Differentiation in HSC Mathematics: What You Need to Know!

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RuiAce:
To visualise a scenario where they give the same result

sophroberts812:

--- Quote from: jamonwindeyer on June 29, 2016, 07:31:04 pm ---Hey soph! Welcome to the forums!  ;D ;D ;D

So basically there will always be a "best rule" for the job. In general:


* The product rule is best for when you have two functions of x multiplied together. EG:
* The quotient rule is best for when you have a fraction with some function of x on the denominator (and potentially the numerator as well). EG:
* The chain rule is the only real rule you can use for functions where you have one function inside another, EG:
You are correct in saying that, in some cases, different rules can work!! The product and quotient rules can be swapped with clever manipulation of indices if you so choose, but this almost always makes it harder. Pretty much, if they are multiplied, use product. If you have a fraction, use quotient. If you have a function inside a function, use the chain rule (nothing else will work easily in this case).

I hope this helps!!  ;D ;D

PS - Let me know if you need any help finding anything on the forums!! Happy to see you around  ;D

--- End quote ---


Thanks so much Jamon! Taught me more in 2 minutes than my maths teacher all year - thank you!!

brenden:

--- Quote from: sophroberts812 on June 30, 2016, 09:09:24 am ---Taught me more in 2 minutes than my maths teacher all year - thank you!!

--- End quote ---
lol

anotherworld2b:
I have a quick question how do you check if the function is odd or even and check for asymptotes (vertical and horizontal)?

jakesilove:

--- Quote from: anotherworld2b on January 21, 2017, 12:37:12 am ---I have a quick question how do you check if the function is odd or even and check for asymptotes (vertical and horizontal)?

--- End quote ---

A function is odd if



And even if



Let's use an example to make this clearer. We know that the standard parabola is even. This has equation



Now, let's prove it is even. We start by turning all positive x's into negative x's



But, remember that a negative squared is a positive



So, we've proved the above relation! Let's try now for a cubic





Now, remember that a negative cubed is still a negative



Therefore, the cubic is clearly odd.

I could by wrong, but in 2U I'm pretty sure you just need vertical asymptotes. These will occur when a denominator will be equal to zero. So, the function



will have an asymptote at x=2, as 2-2=0 (which is obviously not allowed).

Maybe horizontal asymptotes are also in the syllabus? In that case, we utilise limits. Say we had a function



Let's restrict it to the positive x for ease, but note that the function is even so the negative x's will have the same asymptote (can you see why?)
We want to know what happens for very large values of x. We start by dividing through by the highest power of x in the denominator (always do this).



Now, we let x approach infinity. What happens when a number is divided by infinity? Well, it turns into zero! So, we know that



And this is our asymptote

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